Higher-order finite element approximation of the dynamic Laplacian

Abstract

The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in [14]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.

Linear response for the dynamic Laplacian and finite-time coherent sets

“…Note that according to our standing assumption, λ 0 is simple and so λ0 is simple if the elements are fine enough (cf [8], lemma 3.65 and [34]) and the kernel of the matrix K − λ0 M is spanned by ũ0 . Thus, on Ṽ0 , the matrix K − λ0 M is nonsingular and (K, M and L are symmetric)…”

An Efficient Automatic Mesh Generator With Parabolic Arcs in Julia for Computation of TE and TM Modes for Waveguides

“…Recently, natural frequency analysis performed with JuliaFEM was detailed in [23]. Most recently, with an efficient implementation of the Julia package, quadratic elements could give a better asymptotic convergence order than linear elements [24].…”